Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) d σ X$ Recall: Factorization in Soft and Collinear Limits P(z) : “Altarelli-Parisi Splitting Functions” (more later) s C P ( z ) i || j | M ( . . . , p i , p j . . . ) | 2 | M ( . . . , p i + p j , . . . ) | 2 → g 2 s ij “Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later) s C 2 s ik j g → 0 | M ( . . . , p i , p j , p k . . . ) | 2 | M ( . . . , p i , p k , . . . ) | 2 g 2 → QCD s ij s jk Lecture III P . Skands 11
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) d σ X$ ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 d σ X s s i 1 s 1 j Recall: Factorization in Soft and Collinear Limits P(z) : “Altarelli-Parisi Splitting Functions” (more later) s C P ( z ) i || j | M ( . . . , p i , p j . . . ) | 2 | M ( . . . , p i + p j , . . . ) | 2 → g 2 s ij “Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later) s C 2 s ik j g → 0 | M ( . . . , p i , p j , p k . . . ) | 2 | M ( . . . , p i , p k , . . . ) | 2 g 2 → QCD s ij s jk Lecture III P . Skands 11
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & d σ X$ ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j Recall: Factorization in Soft and Collinear Limits P(z) : “Altarelli-Parisi Splitting Functions” (more later) s C P ( z ) i || j | M ( . . . , p i , p j . . . ) | 2 | M ( . . . , p i + p j , . . . ) | 2 → g 2 s ij “Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later) s C 2 s ik j g → 0 | M ( . . . , p i , p j , p k . . . ) | 2 | M ( . . . , p i , p k , . . . ) | 2 g 2 → QCD s ij s jk Lecture III P . Skands 11
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & d σ X$ ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 d σ X +1 s s i 2 s 2 j Recall: Factorization in Soft and Collinear Limits P(z) : “Altarelli-Parisi Splitting Functions” (more later) s C P ( z ) i || j | M ( . . . , p i , p j . . . ) | 2 | M ( . . . , p i + p j , . . . ) | 2 → g 2 s ij “Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later) s C 2 s ik j g → 0 | M ( . . . , p i , p j , p k . . . ) | 2 | M ( . . . , p i , p k , . . . ) | 2 g 2 → QCD s ij s jk Lecture III P . Skands 11
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j Recall: Factorization in Soft and Collinear Limits P(z) : “Altarelli-Parisi Splitting Functions” (more later) s C P ( z ) i || j | M ( . . . , p i , p j . . . ) | 2 | M ( . . . , p i + p j , . . . ) | 2 → g 2 s ij “Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later) s C 2 s ik j g → 0 | M ( . . . , p i , p j , p k . . . ) | 2 | M ( . . . , p i , p k , . . . ) | 2 g 2 → QCD s ij s jk Lecture III P . Skands 11
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Recall: Factorization in Soft and Collinear Limits P(z) : “Altarelli-Parisi Splitting Functions” (more later) s C P ( z ) i || j | M ( . . . , p i , p j . . . ) | 2 | M ( . . . , p i + p j , . . . ) | 2 → g 2 s ij “Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later) s C 2 s ik j g → 0 | M ( . . . , p i , p j , p k . . . ) | 2 | M ( . . . , p i , p k , . . . ) | 2 g 2 → QCD s ij s jk Lecture III P . Skands 11
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Recall: Singularities mandated by gauge theory Non-singular terms: up to you QCD Lecture III P . Skands 12
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Recall: Singularities mandated by gauge theory Non-singular terms: up to you SOFT COLLINEAR 2 s ik ✓ s ij |M ( Z 0 → q i g j ¯ q k ) | 2 ◆� 1 + s jk q K ) | 2 = g 2 s 2 C F + |M ( Z 0 → q I ¯ s ij s jk s IK s jk s ij 2 s ik ✓ s ij |M ( H 0 → q i g j ¯ q k ) | 2 ◆� 1 + s jk q K ) | 2 = g 2 s 2 C F + + 2 |M ( H 0 → q I ¯ QCD s ij s jk s IK s jk s ij Lecture SOFT COLLINEAR +F III P . Skands 12
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Iterated factorization Gives us an approximation to ∞ -order tree-level cross sections. Exact in singular (strongly ordered) limit. Finite terms → Uncertainty on non-singular (hard) radiation QCD Lecture III P . Skands 13
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Iterated factorization Gives us an approximation to ∞ -order tree-level cross sections. Exact in singular (strongly ordered) limit. Finite terms → Uncertainty on non-singular (hard) radiation But something is not right … Total σ would be infinite … QCD Lecture III P . Skands 13
Loops and Legs Coefficients of the Perturbative Series X (2) X+1 (2) … s p o X+1 (1) X+2 (1) X+3 (1) X (1) … o L Universality (scaling) Born X+1 (0) X+2 (0) X+3 (0) … Jet-within-a-jet-within-a-jet-... L e g s QCD Lecture III P . Skands 14
Loops and Legs Coefficients of the Perturbative Series The corrections from X (2) X+1 (2) … Quantum Loops are missing s p o X+1 (1) X+2 (1) X+3 (1) X (1) … o L Universality (scaling) Born X+1 (0) X+2 (0) X+3 (0) … Jet-within-a-jet-within-a-jet-... L e g s QCD Lecture III P . Skands 14
The Resummation Idea d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j ► Interpretation: the structure evolves! (example: X = 2-jets) • Take a jet algorithm, with resolution measure “Q”, apply it to your events • At a very crude resolution, you find that everything is 2-jets • At finer resolutions some 2-jets migrate 3-jets = σ X+1 (Q) = σ X;incl – σ X;excl (Q) • Later, some 3-jets migrate further, etc σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) • This evolution takes place between two scales, Q in ~ s and Q end ~ Q had ► σ X;tot = Sum ( σ X+0,1,2,3,…;excl ) = int( d σ X ) QCD Lecture III P . Skands 15
The Resummation Idea d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j ► Interpretation: the structure evolves! (example: X = 2-jets) • Take a jet algorithm, with resolution measure “Q”, apply it to your events • At a very crude resolution, you find that everything is 2-jets • At finer resolutions some 2-jets migrate 3-jets = σ X+1 (Q) = σ X;incl – σ X;excl (Q) • Later, some 3-jets migrate further, etc σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) • This evolution takes place between two scales, Q in ~ s and Q end ~ Q had ► σ X;tot = Sum ( σ X+0,1,2,3,…;excl ) = int( d σ X ) QCD Lecture III P . Skands 15
The Resummation Idea d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j ► Interpretation: the structure evolves! (example: X = 2-jets) • Take a jet algorithm, with resolution measure “Q”, apply it to your events • At a very crude resolution, you find that everything is 2-jets • At finer resolutions some 2-jets migrate 3-jets = σ X+1 (Q) = σ X;incl – σ X;excl (Q) • Later, some 3-jets migrate further, etc σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) • This evolution takes place between two scales, Q in ~ s and Q end ~ Q had ► σ X;tot = Sum ( σ X+0,1,2,3,…;excl ) = int( d σ X ) QCD Lecture III P . Skands 15
The Resummation Idea d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j ► Interpretation: the structure evolves! (example: X = 2-jets) • Take a jet algorithm, with resolution measure “Q”, apply it to your events • At a very crude resolution, you find that everything is 2-jets • At finer resolutions some 2-jets migrate 3-jets = σ X+1 (Q) = σ X;incl – σ X;excl (Q) • Later, some 3-jets migrate further, etc σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) • This evolution takes place between two scales, Q in ~ s and Q end ~ Q had ► σ X;tot = Sum ( σ X+0,1,2,3,…;excl ) = int( d σ X ) QCD Lecture III P . Skands 15
The Resummation Idea d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j ► Interpretation: the structure evolves! (example: X = 2-jets) • Take a jet algorithm, with resolution measure “Q”, apply it to your events • At a very crude resolution, you find that everything is 2-jets • At finer resolutions some 2-jets migrate 3-jets = σ X+1 (Q) = σ X;incl – σ X;excl (Q) • Later, some 3-jets migrate further, etc σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) • This evolution takes place between two scales, Q in ~ s and Q end ~ Q had ► σ X;tot = Sum ( σ X+0,1,2,3,…;excl ) = int( d σ X ) QCD Lecture III P . Skands 15
Evolution Q ∼ Q X Leading Order “Experiment” 100 100 75 75 % 50 50 25 25 0 0 Born +1 +2 Born (exc) +1 (exc) +2 (inc) QCD Lecture III P . Skands 16
Evolution Q X Q ∼ “A few” Leading Order “Experiment” 100 100 75 75 % 50 50 25 25 0 0 Born +1 +2 Born (exc) +1 (exc) +2 (inc) QCD Lecture III P . Skands 17
Evolution Q ⌧ Q X Leading Order “Experiment” 400 100 300 75 % 200 50 100 25 0 0 Born + 1 + 2 Born (exc) + 1 (exc) + 2 (inc) QCD Cross Section Remains = Born (IR safe) Cross Section Diverges Number of Partons Diverges (IR unsafe) Lecture III P . Skands 18
Evolution Equations What we need is a differential equation Boundary condition: a few partons defined at a high scale (Q F ) Then evolves (or “runs”) that parton system down to a low scale (the hadronization cutoff ~ 1 GeV) → It’s an evolution equation in Q F QCD Lecture III P . Skands 19
Evolution Equations What we need is a differential equation Boundary condition: a few partons defined at a high scale (Q F ) Then evolves (or “runs”) that parton system down to a low scale (the hadronization cutoff ~ 1 GeV) → It’s an evolution equation in Q F Close analogue: nuclear decay Evolve an unstable nucleus. Check if it decays + follow chains of decays. Decay constant Probability to remain undecayed in the time interval [ t 1 , t 2 ] Z t 2 d P ( t ) ✓ ◆ = c N ∆ ( t 1 , t 2 ) = exp c N d t = exp ( − c N ∆ t ) − d t t 1 = 1 − c N ∆ t + O ( c 2 N ) Decay probability per unit time d P res ( t ) = − d ∆ = c N ∆ ( t 1 , t ) d t d t ∆ (t 1 ,t 2 ) : “Sudakov Factor” QCD (requires that the nucleus did not already decay) Lecture III P . Skands 19
Nuclear Decay � | | � t 2 � � d t d P Nuclei remaining undecayed ∆ ( t 1 , t 2 ) = exp = after time t − d t t 1 100 % Second Order 50 % All Orders Exponential Time 0 % Early Times Late Times Third Order First Order -50 % QCD -100 % Lecture III P . Skands 20
The Sudakov Factor In nuclear decay, the Sudakov factor counts: How many nuclei remain undecayed after a time t Probability to remain undecayed in the time interval [ t 1 , t 2 ] Z t 2 ✓ ◆ ∆ ( t 1 , t 2 ) = exp c N d t = exp ( − c N ∆ t ) − t 1 QCD Lecture III P . Skands 21
The Sudakov Factor In nuclear decay, the Sudakov factor counts: How many nuclei remain undecayed after a time t Probability to remain undecayed in the time interval [ t 1 , t 2 ] Z t 2 ✓ ◆ ∆ ( t 1 , t 2 ) = exp c N d t = exp ( − c N ∆ t ) − t 1 In parton showers, we may also define a Sudakov factor for the parton system. It counts The probability that the parton system doesn’t evolve (emit) when I run the factorization scale (~1/time) from a high to a lower scale Evolution probability per unit time (replace c N by proper shower evolution kernels) d P res ( t ) = − d ∆ = c N ∆ ( t 1 , t ) d t d t QCD Lecture III P . Skands 21
What’s the evolution kernel? Altarelli-Parisi splitting functions Can be derived ( in the collinear limit ) from requiring invariance of the physical result with respect to Q F → RGE Altarelli-Parisi 1 + z 2 P (E.g., PYTHIA) P q → qg ( z ) = C F 1 − z , α abc (1 − z (1 − z )) 2 � d P a = 2 π P a → bc ( z ) d t d z . P g → gg ( z ) = N C , z (1 − z ) b,c T R ( z 2 + (1 − z ) 2 ) , P g → qq ( z ) = c 1 + z 2 b e 2 P q → q � ( z ) = 1 − z , a q p b = z p a 1 + z 2 e 2 P ⇥ → ⇥� ( z ) = 1 − z , p c = (1-z) p a ⇥ … with Q 2 some measure of event/jet resolution d t = d Q 2 measuring parton virtualities / formation time / … Q 2 = d ln Q 2 QCD Different models make different choices But choice is not entirely free … Lecture III P . Skands 22
Coherence Illustrations by T. Sjöstrand QED: Chudakov effect (mid-fifties) e + e − cosmic ray γ atom reduced normal emulsion plate ionization ionization QCD: colour coherence for soft gluon emission 2 2 = + → an example of an interference effect that can be treated probabilistically solved by requiring emission angles to be decreasing QCD More interference effects can be included by matching to full matrix elements → tomorrow Lecture III P . Skands 23
Coherence Illustrations by T. Sjöstrand QED: Chudakov effect (mid-fifties) Approximations to e + Coherence: e − cosmic ray γ atom Angular Ordering (HERWIG) Angular Vetos (PYTHIA) Coherent Dipoles/Antennae (ARIADNE, CS, VINCIA) reduced normal emulsion plate ionization ionization QCD: colour coherence for soft gluon emission 2 2 = + → an example of an interference effect that can be treated probabilistically solved by requiring emission angles to be decreasing QCD More interference effects can be included by matching to full matrix elements → tomorrow Lecture III P . Skands 23
What is t ? t : Shower Evolution Measure ~ Jet Resolution Measure ~ Sliding Factorization Scale d t = d Q 2 Q 2 = d ln Q 2 s ij s jk P HASE S PACE FOR 2 → 3 KINEMATICS INCLUDING (E, P ) CONS Original Dipole-Antenna: I K ¯ q q Collinear with ¯ q Collinear with K j i k QCD Collinear with I Collinear with q s ij s jk Soft Soft Lecture III P . Skands 24
What is t ? t : Shower Evolution Measure Parton Showers (PYTHIA & HERWIG) ~ Jet Resolution Measure 2 Virtuality � Ordering: side a p Tevol � Ordering: side a Angular Ordering ~ Sliding Factorization Scale 1.0 1.0 1.0 2 Virtuality � Ordering: side b p Tevol � Ordering: side b 1.0 1.0 0.8 0.8 y rb � s rb � s arb � 1 � x a y rb � s rb � s arb � 1 � x a 0.8 0.8 0.8 y rb � s rb � s arb � 1 � x a 0.6 y rb � s rb � s arb � 1 � x a 0.6 y rb � s rb � s arb � 1 � x a 0.25 0.4 0.4 0.25 d t = d Q 2 0.6 0.2 0.6 0.2 0.6 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Q 2 = d ln Q 2 y ar � s ar � s arb � 1 � x b y ar � s ar � s arb � 1 � x b 0.4 0.4 0.4 0.5 0.2 0.2 0.2 0.75 0.75 0.5 0.25 0.5 0.75 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y ar � s ar � s arb � 1 � x b y ar � s ar � s arb � 1 � x b y ar � s ar � s arb � 1 � x b s ij s jk P HASE S PACE FOR 2 → 3 PYTHIA: imposes angular vetos to obtain coherence KINEMATICS INCLUDING (E, P ) CONS HERWIG:coherent (by angular ordering) but has dead zone Original Dipole-Antenna: I K ¯ q q Collinear with ¯ q Collinear with K j i k QCD Collinear with I Collinear with q s ij s jk Soft Soft Lecture III P . Skands 24
What is t ? t : Shower Evolution Measure Parton Showers (PYTHIA & HERWIG) ~ Jet Resolution Measure 2 Virtuality � Ordering: side a p Tevol � Ordering: side a Angular Ordering ~ Sliding Factorization Scale 1.0 1.0 1.0 2 Virtuality � Ordering: side b p Tevol � Ordering: side b 1.0 1.0 0.8 0.8 y rb � s rb � s arb � 1 � x a y rb � s rb � s arb � 1 � x a 0.8 0.8 0.8 y rb � s rb � s arb � 1 � x a 0.6 y rb � s rb � s arb � 1 � x a 0.6 y rb � s rb � s arb � 1 � x a 0.25 0.4 0.4 0.25 d t = d Q 2 0.6 0.2 0.6 0.2 0.6 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Q 2 = d ln Q 2 y ar � s ar � s arb � 1 � x b y ar � s ar � s arb � 1 � x b 0.4 0.4 0.4 0.5 0.2 0.2 0.2 0.75 0.75 0.5 0.25 0.5 0.75 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 y ar � s ar � s arb � 1 � x b y ar � s ar � s arb � 1 � x b y ar � s ar � s arb � 1 � x b s ij s jk P HASE S PACE FOR 2 → 3 PYTHIA: imposes angular vetos to obtain coherence KINEMATICS INCLUDING (E, P ) CONS HERWIG:coherent (by angular ordering) but has dead zone Original Dipole-Antenna: I K ¯ q q Collinear with ¯ q Collinear with K Dipole/Antenna Showers (ARIADNE, SHERPA, VINCIA) Mass-Ordering p ⊥ -ordering ( m 2 ⌦ m 2 ↵ min ) ( geometric ) ( j 1.0 1.0 0.4 0.2 0.8 0.8 Linear in y 0.6 0.6 0.8 0.4 y jk y jk i k 0.4 0.8 0.4 0.8 0.6 0.6 0.2 0.4 0.2 0.2 0.2 0.6 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 QCD y ij y ij (a) Q 2 E = m 2 (b) Q 2 D = 2 min( y ij , y jk ) s E = 2 p ⊥ √ s = 2 √ y ij y jk s Collinear with I Collinear with q s ij s jk Soft Soft Lecture .0 .0 Intrinsically Coherent III .6 .8 .8 .2 .6 .6 .6 .8 P . Skands 24 jk jk .8 .4 .4 .4 .6 .8 .4 .2 .2 .2 .2 .4 .0 .0 .0 .2 .4 .6 0.8 1.0 .0 .2 .4 .6 .8 1.0 y y
Antennae Observation: the evolution kernel is responsible for generating real radiation. → Choose it to be the ratio of the real-emission matrix element to the Born-level matrix element → AP in coll limit, but also includes the Eikonal for soft radiation. ⇥ 2 → 3 instead of 1 → 2 Dipole-Antennae ( → all partons on shell) (E.g., ARIADNE, VINCIA) d s ij d s jk d P IK → ijk = 16 π 2 s a ( s ij , s jk ) ⇤ QCD Lecture III P . Skands 25
Antennae Observation: the evolution kernel is responsible for generating real radiation. → Choose it to be the ratio of the real-emission matrix element to the Born-level matrix element → AP in coll limit, but also includes the Eikonal for soft radiation. ⇥ 2 → 3 instead of 1 → 2 Dipole-Antennae ( → all partons on shell) (E.g., ARIADNE, VINCIA) d s ij d s jk d P IK → ijk = 16 π 2 s a ( s ij , s jk ) ⇤ s QCD Lecture III P . Skands 25
Antennae Observation: the evolution kernel is responsible for generating real radiation. → Choose it to be the ratio of the real-emission matrix element to the Born-level matrix element → AP in coll limit, but also includes the Eikonal for soft radiation. ⇥ 2 → 3 instead of 1 → 2 Dipole-Antennae ( → all partons on shell) (E.g., ARIADNE, VINCIA) d s ij d s jk d P IK → ijk = 16 π 2 s a ( s ij , s jk ) ⇤ i I j s k K (s ij ,s jk ) (…) QCD Lecture (…) III P . Skands 25
Antennae Observation: the evolution kernel is responsible for generating real radiation. → Choose it to be the ratio of the real-emission matrix element to the Born-level matrix element → AP in coll limit, but also includes the Eikonal for soft radiation. ⇥ 2 → 3 instead of 1 → 2 Dipole-Antennae ( → all partons on shell) (E.g., ARIADNE, VINCIA) d s ij d s jk d P IK → ijk = 16 π 2 s a ( s ij , s jk ) q = 2 C F � ⇥ 2 s ik s + s 2 ij + s 2 a q ¯ q → qg ¯ jk s ij s jk ⇤ C A � ⇥ 2 s ik s + s 2 ij + s 2 jk − s 3 a qg → qgg = ij s ij s jk i C A � 2 s ik s + s 2 ij + s 2 jk − s 3 ij − s 3 ⇥ a gg → ggg = I jk s ij s jk j s q 0 q 0 = T R � s − 2 s ij + 2 s 2 ⇥ k K a qg → q ¯ ij s jk q 0 q 0 = a qg → q ¯ (s ij ,s jk ) (…) QCD a gg → g ¯ q 0 q 0 … + non-singular terms Lecture (…) III P . Skands 25
Evolution → Unitarity ✓ d σ X = For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 . . . d σ X +1 s s i 2 s 2 j Imposed by Event evolution : Unitarity When (X) branches to (X+1): Gain one (X+1). Loose one (X). Kinoshita-Lee-Nauenberg: → evolution equation with kernel d σ X +1 Loop = - Int(Tree) + F d σ X Neglect F → Leading-Logarithmic (LL) Evolve in some measure of resolution Approximation ~ virtuality, energy, … ~ fractal scale QCD → includes both real (tree) and virtual (loop) corrections Lecture III P . Skands 26
Bootstrapped Perturbation Theory Resummation Born + Shower X (2) X+1 (2) … Unitarity s p o X+1 (1) X+2 (1) X+3 (1) X (1) … o L Exponentiation Universality (scaling) Born X+1 (0) X+2 (0) X+3 (0) … Jet-within-a-jet-within-a-jet-... L e g s QCD Lecture III P . Skands 27
Bootstrapped Perturbation Theory Resummation Born + Shower X (2) X+1 (2) … Unitarity s p o X+1 (1) X+2 (1) X+3 (1) X (1) … o L Exponentiation Universality (scaling) Born X+1 (0) X+2 (0) X+3 (0) … Jet-within-a-jet-within-a-jet-... L e g s QCD → lecture on Matching Lecture III P . Skands 27
The Shower Operator d σ H � H = Hard process � H | 2 δ ( O − O ( { p } H )) d Φ H | M (0) Born � = � d O {p} : partons � Born But instead of evaluating O directly on the Born final state, first insert a showering operator QCD Lecture III P . Skands 28
The Shower Operator d σ H � H = Hard process � H | 2 δ ( O − O ( { p } H )) d Φ H | M (0) Born � = � d O {p} : partons � Born But instead of evaluating O directly on the Born final state, first insert a showering operator Born {p} : partons d σ H � � H | 2 S ( { p } H , O ) d Φ H | M (0) � = + shower S : showering operator � d O � S r — the evolution operator — will be responsib QCD Lecture III P . Skands 28
The Shower Operator d σ H � H = Hard process � H | 2 δ ( O − O ( { p } H )) d Φ H | M (0) Born � = � d O {p} : partons � Born But instead of evaluating O directly on the Born final state, first insert a showering operator Born {p} : partons d σ H � � H | 2 S ( { p } H , O ) d Φ H | M (0) � = + shower S : showering operator � d O � S r — the evolution operator — will be responsib Unitarity: to first order, S does nothing S ( { p } H , O ) = δ ( O − O ( { p } H )) + O ( α s ) QCD Lecture III P . Skands 28
The Shower Operator To ALL Orders S ( { p } X , O ) = ∆ ( t start , t had ) δ ( O − O ( { p } X )) � → “Nothing Happens” “Evaluate Observable” � t had d t d ∆ ( t start , t ) )) − S ( { p } X +1 , O ) d t t start → “Something Happens” “Continue Shower” � All-orders Probability that nothing happens | | � t 2 (Exponentiation) � d t d P � ∆ ( t 1 , t 2 ) = exp Analogous to nuclear decay − d t N(t) ≈ N(0) exp(-ct) t 1 QCD Lecture III P . Skands 29
The Shower Operator To ALL Orders (Markov Chain) S ( { p } X , O ) = ∆ ( t start , t had ) δ ( O − O ( { p } X )) � → “Nothing Happens” “Evaluate Observable” � t had d t d ∆ ( t start , t ) )) − S ( { p } X +1 , O ) d t t start → “Something Happens” “Continue Shower” � All-orders Probability that nothing happens | | � t 2 (Exponentiation) � d t d P � ∆ ( t 1 , t 2 ) = exp Analogous to nuclear decay − d t N(t) ≈ N(0) exp(-ct) t 1 QCD Lecture III P . Skands 29
A Shower Algorithm 1. Generate Random Number, R ∈ [0,1] 1.0 0.8 y jk � s jk � s ijk � 1 � x i 0.6 Solve equation for t (with starting scale t 1 ) R = ∆ ( t 1 , t ) 0.4 Analytically for simple splitting kernels, else numerically (or by trial+veto) 0.2 0.0 → t scale for next branching 0.0 0.2 0.4 0.6 0.8 1.0 y ij � s ij � s ijk � 1 � x k QCD Lecture III P . Skands 30
A Shower Algorithm t 1. Generate Random Number, R ∈ [0,1] 1.0 0.8 y jk � s jk � s ijk � 1 � x i t 1 0.6 Solve equation for t (with starting scale t 1 ) R = ∆ ( t 1 , t ) 0.4 Analytically for simple splitting kernels, else numerically (or by trial+veto) 0.2 0.0 → t scale for next branching 0.0 0.2 0.4 0.6 0.8 1.0 y ij � s ij � s ijk � 1 � x k QCD Lecture III P . Skands 30
A Shower Algorithm t 1. Generate Random Number, R ∈ [0,1] 1.0 0.8 y jk � s jk � s ijk � 1 � x i t 1 0.6 Solve equation for t (with starting scale t 1 ) R = ∆ ( t 1 , t ) 0.4 Analytically for simple splitting kernels, else numerically (or by trial+veto) 0.2 0.0 → t scale for next branching 0.0 0.2 0.4 0.6 0.8 1.0 y ij � s ij � s ijk � 1 � x k 2. Generate another Random Number, R z ∈ [0,1] To find second (linearly independent) phase-space invariant I z ( z, t ) Solve equation for z (at scale t ) R z = I z ( z max ( t ) , t ) Z z � d z d ∆ ( t 0 ) With the “primitive function” � I z ( z, t ) = � d t 0 � z min ( t ) t 0 = t QCD Lecture III P . Skands 30
A Shower Algorithm t 1. Generate Random Number, R ∈ [0,1] 1.0 0.8 y jk � s jk � s ijk � 1 � x i t 1 0.6 Solve equation for t (with starting scale t 1 ) R = ∆ ( t 1 , t ) 0.4 (t,z) Analytically for simple splitting kernels, else numerically (or by trial+veto) 0.2 0.0 → t scale for next branching 0.0 0.2 0.4 0.6 0.8 1.0 y ij � s ij � s ijk � 1 � x k 2. Generate another Random Number, R z ∈ [0,1] To find second (linearly independent) phase-space invariant I z ( z, t ) Solve equation for z (at scale t ) R z = I z ( z max ( t ) , t ) Z z � d z d ∆ ( t 0 ) With the “primitive function” � I z ( z, t ) = � d t 0 � z min ( t ) t 0 = t QCD Lecture III P . Skands 30
A Shower Algorithm t 1. Generate Random Number, R ∈ [0,1] 1.0 0.8 y jk � s jk � s ijk � 1 � x i t 1 0.6 Solve equation for t (with starting scale t 1 ) R = ∆ ( t 1 , t ) 0.4 (t,z) Analytically for simple splitting kernels, else numerically (or by trial+veto) 0.2 0.0 → t scale for next branching 0.0 0.2 0.4 0.6 0.8 1.0 y ij � s ij � s ijk � 1 � x k 2. Generate another Random Number, R z ∈ [0,1] To find second (linearly independent) phase-space invariant I z ( z, t ) Solve equation for z (at scale t ) R z = I z ( z max ( t ) , t ) Z z � d z d ∆ ( t 0 ) With the “primitive function” � I z ( z, t ) = � d t 0 � z min ( t ) t 0 = t 3. Generate a third Random Number, R φ ∈ [0,1] QCD Solve equation for φ → Can now do 3D branching R ϕ = ϕ / 2 π Lecture III P . Skands 30
Ambiguities The final states generated by the shower algorithm will depend on 1. The choice of perturbative evolution variable(s) t [ i ] . 2. The choice of phase-space mapping d Φ [ i ] n +1 / d Φ n . 3. The choice of radiation functions a i , as a function of the phase-space variables. 4. The choice of renormalization scale function µ R . 5. Choices of starting and ending scales. QCD Lecture III P . Skands 31
Ambiguities The final states generated by the shower algorithm will depend on 1. The choice of perturbative evolution variable(s) t [ i ] . 2. The choice of phase-space mapping d Φ [ i ] n +1 / d Φ n . 3. The choice of radiation functions a i , as a function of the phase-space variables. 4. The choice of renormalization scale function µ R . 5. Choices of starting and ending scales. → gives us additional handles for uncertainty estimates, beyond just μ R QCD Lecture III P . Skands 31
(Physics Consequences) Subleading Issues Hard Jet Substructure (showers approximate 1 → 3 by iterated 1 → 2, but full 1 → 3 kernels have additional structure. Iterated 1 → 2 only works when successive emissions are strongly ordered (dominant) but not when two or more emissions happen at ~ the same scale → hard substructure) p T kicks from recoil strategy (global vs local; 1 → 2 vs 2 → 3) _ Gluon Splittings g → qq (less well controlled than gluon emission) Mass Effects (example: b-jet calibration vs light-jet) Subleading coherence (e.g., angular-ordered parton showers vs p T - ordered dipole ones, in particular initial-final connections…) QCD Lecture III P . Skands 32
(Physics Consequences) Subleading Issues Hard Jet Substructure (showers approximate 1 → 3 by iterated 1 → 2, but full 1 → 3 kernels have additional structure. Iterated 1 → 2 only works when successive emissions are strongly ordered (dominant) but not when two or more emissions happen at ~ the same scale → hard substructure) p T kicks from recoil strategy (global vs local; 1 → 2 vs 2 → 3) _ Gluon Splittings g → qq (less well controlled than gluon emission) Current “holy grail”: Include full higher-order splitting kernels Mass Effects (example: b-jet calibration vs light-jet) → will reduce all these ambiguities Active field of research. Subleading coherence (e.g., angular-ordered parton showers vs p T - For now, must do our best to estimate ordered dipole ones, in particular initial-final connections…) QCD the uncertainties. Lecture III P . Skands 32
Tuning 1. Fragmentation Tuning Perturbative: jet radiation, jet broadening, jet structure Non-perturbative: hadronization modeling & parameters 2. Initial-State Tuning Perturbative: initial-state radiation, initial-final interference Non-perturbative: PDFs, primordial k T 3. Underlying-Event & Min-Bias Tuning Perturbative: Multi-parton interactions, rescattering Non-perturbative: Multi-parton PDFs, Beam Remnant fragmentation, Color (re)connections, collective effects, impact parameter dependence, … QCD Lecture III 33 P . Skands
Example: pQCD Shower Tuning Main pQCD Parameters α s (m Z ) The value of the strong coupling at the Z pole Governs overall amount of radiation α s Running Renormalization Scheme and Scale for α s 1- / 2-loop running, MSbar / CMW scheme, μ R ~ Q 2 or p T2 Matching Additional Matrix Elements included? At tree level / one-loop level? Using what scheme? S u b l e a d i n g L o g s Ordering variable, coherence treatment, effective 1 → 3 (or 2 → 4), recoil strategy, etc QCD Lecture V P . Skands 34
Need IR Corrections? PYTHIA 8 (hadronization off) vs LEP: Thrust Major �� i | � p i · � n | � 1 − T → 1 T = max � i | � p i | 2 n � 1 − T → 0 Minor 1/N dN/d(1-T) 1/N dN/d(Major) 1/N dN/d(Minor) 1/N dN/d(O) 1-Thrust (udsc) Major Minor Oblateness Delphi Delphi Delphi L3 10 10 10 10 Pythia Pythia Pythia Pythia 1 1 1 1 -1 -1 -1 -1 Oblateness 10 10 10 10 Minor 1-T Major = Major - Minor -2 -2 -2 -2 10 10 10 10 V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T Data from Phys.Rept. 399 (2004) 71 Data from CERN-PPE-96-120 Data from CERN-PPE-96-120 Data from CERN-PPE-96-120 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 10 10 10 10 1.4 1.4 1.4 1.4 Theory/Data Theory/Data Theory/Data Theory/Data 1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 1-T (udsc) Major Minor O Significant Discrepancies (>10%) QCD for T < 0.05, Major < 0.15, Minor < 0.2, and for all values of Oblateness Lecture III P . Skands 35
Need IR Corrections? PYTHIA 8 (hadronization on) vs LEP: Thrust Major �� i | � p i · � n | � 1 − T → 1 T = max � i | � p i | 2 n � 1 − T → 0 Minor 1/N dN/d(1-T) 1/N dN/d(Minor) 1/N dN/d(Major) 1/N dN/d(O) 1-Thrust (udsc) Major Minor Oblateness Delphi L3 Delphi Delphi 10 10 10 10 Pythia Pythia Pythia Pythia 1 1 1 1 -1 -1 -1 -1 10 10 10 10 -2 -2 -2 -2 10 10 10 10 V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T Data from Phys.Rept. 399 (2004) 71 Data from CERN-PPE-96-120 Data from CERN-PPE-96-120 Data from CERN-PPE-96-120 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 10 10 10 10 1.4 1.4 1.4 1.4 Theory/Data Theory/Data Theory/Data Theory/Data 1.2 1.2 1.2 1.2 1 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0 0.2 0.4 0.6 1-T (udsc) Major Minor O QCD Lecture III P . Skands 36
Need IR Corrections? PYTHIA 8 (hadronization on) vs LEP: Thrust Major �� i | � p i · � n | � 1 − T → 1 T = max � i | � p i | 2 n � 1 − T → 0 Minor 1/N dN/d(1-T) 1/N dN/d(Minor) 1/N dN/d(Major) 1/N dN/d(O) 1-Thrust (udsc) Major Minor Oblateness Delphi L3 Delphi Delphi 10 10 10 10 Pythia Pythia Pythia Pythia 1 1 1 1 -1 -1 -1 -1 10 10 10 10 -2 -2 -2 -2 10 10 10 10 V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T Data from Phys.Rept. 399 (2004) 71 Data from CERN-PPE-96-120 Data from CERN-PPE-96-120 Data from CERN-PPE-96-120 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 10 10 10 10 1.4 1.4 1.4 1.4 Theory/Data Theory/Data Theory/Data Theory/Data 1.2 1.2 1.2 1.2 1 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0 0.2 0.4 0.6 1-T (udsc) Major Minor O Note: Value of Strong coupling is α s (M Z ) = 0.14 QCD Lecture III P . Skands 36
Value of Strong Coupling PYTHIA 8 (hadronization on) vs LEP: Thrust Major �� i | � p i · � n | � 1 − T → 1 T = max � i | � p i | 2 � n 1 − T → 0 Minor 1/N dN/d(Minor) 1/N dN/d(1-T) 1/N dN/d(Major) 1/N dN/d(O) 1-Thrust (udsc) Major Minor Oblateness Delphi Delphi Delphi L3 10 10 10 10 Pythia Pythia Pythia Pythia 1 1 1 1 -1 -1 -1 -1 10 10 10 10 -2 -2 -2 -2 10 10 10 10 V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T V I N C I A R O O T Data from CERN-PPE-96-120 Data from Phys.Rept. 399 (2004) 71 Data from CERN-PPE-96-120 Data from CERN-PPE-96-120 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 -3 Pythia 8.165 10 10 10 10 1.4 1.4 1.4 1.4 Theory/Data Theory/Data Theory/Data Theory/Data 1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0 0.2 0.4 0.6 1-T (udsc) Major Minor O Note: Value of Strong coupling is α s (M Z ) = 0.12 QCD Lecture III P . Skands 37
Wait … is this Crazy? Best result Obtained with α s (M Z ) ≈ 0.14 ≠ World Average = 0.1176 ± 0.0020 QCD Lecture III P . Skands 38
Wait … is this Crazy? Best result Obtained with α s (M Z ) ≈ 0.14 ≠ World Average = 0.1176 ± 0.0020 Value of α s Depends on the order and scheme MC ≈ Leading Order + LL resummation Other leading-Order extractions of α s ≈ 0.13 - 0.14 Effective scheme interpreted as “CMW” → 0.13; 2-loop running → 0.127; NLO → 0.12 ? QCD Lecture III P . Skands 38
Wait … is this Crazy? Best result Obtained with α s (M Z ) ≈ 0.14 ≠ World Average = 0.1176 ± 0.0020 Value of α s Depends on the order and scheme MC ≈ Leading Order + LL resummation Other leading-Order extractions of α s ≈ 0.13 - 0.14 Effective scheme interpreted as “CMW” → 0.13; 2-loop running → 0.127; NLO → 0.12 ? Not so crazy Tune/measure even pQCD parameters with the actual generator. Sanity check = consistency with other determinations at a similar formal order, within the uncertainty at that order (including a CMW-like scheme redefinition to go to ‘MC scheme’) QCD Lecture III P . Skands 38
Wait … is this Crazy? Best result Obtained with α s (M Z ) ≈ 0.14 ≠ World Average = 0.1176 ± 0.0020 Value of α s Depends on the order and scheme MC ≈ Leading Order + LL resummation Other leading-Order extractions of α s ≈ 0.13 - 0.14 Effective scheme interpreted as “CMW” → 0.13; 2-loop running → 0.127; NLO → 0.12 ? Not so crazy Tune/measure even pQCD parameters with the actual generator. Sanity check = consistency with other determinations at a similar formal order, within the uncertainty at that order (including a CMW-like scheme redefinition to go to ‘MC scheme’) Improve → Matching at LO and NLO QCD Non-perturbative → Lecture on IR Lecture III P . Skands 38
Uncertainties
The Tyranny of Carlo J. D. Bjorken “ Another change that I find disturbing is the rising tyranny of Carlo. No, I don’t mean that fellow who runs CERN, but the other one, with first name Monte. QCD Lecture III P . Skands 40
The Tyranny of Carlo J. D. Bjorken “ Another change that I find disturbing is the rising tyranny of Carlo. No, I don’t mean that fellow who runs CERN, but the other one, with first name Monte. The simultaneous increase in detector complexity and in computation power has made simulation techniques an essential feature of contemporary experimentation. The Monte Carlo simulation has become the major means of visualization of not only detector performance but also of physics phenomena. So far so good. QCD Lecture III P . Skands 40
The Tyranny of Carlo J. D. Bjorken “ Another change that I find disturbing is the rising tyranny of Carlo. No, I don’t mean that fellow who runs CERN, but the other one, with first name Monte. The simultaneous increase in detector complexity and in computation power has made simulation techniques an essential feature of contemporary experimentation. The Monte Carlo simulation has become the major means of visualization of not only detector performance but also of physics phenomena. So far so good. But it often happens that the physics simulations provided by the the MC generators carry the authority of data itself. They look like data and feel like data, and if one is not careful they are accepted as if they were data. All Monte Carlo codes come with a GIGO (garbage in, garbage out) warning label. But the GIGO warning label is just as easy for a physicist to ignore as that little message on a packet of cigarettes is for a chain smoker to ignore. I see nowadays experimental papers that claim agreement with QCD (translation: someone’s simulation labeled QCD) and/or disagreement with an alternative piece of physics (translation: an unrealistic simulation), without much evidence of the inputs into those simulations.” QCD Lecture III P . Skands 40
The Tyranny of Carlo J. D. Bjorken “ Another change that I find disturbing is the rising tyranny of Carlo. No, I don’t mean that fellow who runs CERN, but the other one, with first name Monte. The simultaneous increase in detector complexity and in computation power has made simulation techniques an essential feature of contemporary experimentation. The Monte Carlo simulation has become the major means of visualization of not only detector performance but also of physics phenomena. So far so good. But it often happens that the physics simulations provided by the the MC generators carry the authority of data itself. They look like data and feel like data, and if one is not careful they are accepted as if they were data. All Monte Carlo codes come with a GIGO (garbage in, garbage out) warning label. But the GIGO warning label is just as easy for a physicist to ignore as that little message on a packet of cigarettes is for a chain smoker to ignore. I see nowadays experimental papers that claim agreement with QCD (translation: someone’s simulation labeled QCD) and/or disagreement with an alternative piece of physics (translation: an unrealistic simulation), without much evidence of the inputs into those simulations.” Account for parameters + pertinent cross-checks and validations QCD Do serious effort to estimate uncertainties, by salient variations Lecture III P . Skands 40
Uncertainty Estimates a) Authors provide specific “tune variations” b) One shower run Run once for each variation → envelope + unitarity-based uncertainties → envelope PS, Phys. Rev. D82 (2010) 074018 Giele, Kosower, PS; Phys. Rev. D84 (2011) 054003 1/N dN/d(1-T) 1-Thrust (udsc) 10 L3 Vincia 1 -1 10 VINCIA + PYTHIA 8 example Vincia:uncertaintyBands = on PYTHIA 6 example -2 10 Perugia Variations Vincia 1.027 + MadGraph 4.426 + Pythia 8.153 μ R , K MPI , CR, E cm -scaling, PDFs Data from Phys.Rept. 399 (2004) 71 -3 10 Rel.Unc. 1 0 2 R 1/N Def µ Finite QMatch Ord C 1.4 Theory/Data 1.2 1 0.8 Plot from mcplots.cern.ch 0.6 QCD 0 0.1 0.2 0.3 0.4 0.5 Lecture 1-T (udsc) III P . Skands 41
Uncertainty Estimates a) Authors provide specific “tune variations” b) One shower run Run once for each variation → envelope + unitarity-based uncertainties → envelope PS, Phys. Rev. D82 (2010) 074018 Giele, Kosower, PS; Phys. Rev. D84 (2011) 054003 1/N dN/d(1-T) 1-Thrust (udsc) 10 L3 Vincia 1 -1 10 VINCIA + PYTHIA 8 example Vincia:uncertaintyBands = on PYTHIA 6 example -2 10 Perugia Variations Vincia 1.027 + MadGraph 4.426 + Pythia 8.153 μ R , K MPI , CR, E cm -scaling, PDFs Data from Phys.Rept. 399 (2004) 71 -3 10 Rel.Unc. 1 0 0.1 0.2 0.3 0.4 0.5 0 2 R 1/N Def µ Finite QMatch Ord C 0 0.1 0.2 0.3 0.4 0.5 1.4 Theory/Data 1.2 1 0.8 Matching reduces uncertainty Plot from mcplots.cern.ch 0.6 QCD 0 0.1 0.2 0.3 0.4 0.5 Lecture 1-T (udsc) III P . Skands 42
Automatic Uncertainty Estimates One shower run (VINCIA + PYTHIA) + unitarity-based uncertainties → envelope Giele, Kosower, PS; Phys. Rev. D84 (2011) 054003 *------- PYTHIA Event and Cross Section Statistics -------------------------------------------------------------* | | | Subprocess Code | Number of events | sigma +- delta | | | Tried Selected Accepted | (estimated) (mb) | | | | | |-----------------------------------------------------------------------------------------------------------------| | | | | | f fbar -> gamma*/Z0 221 | 10511 10000 10000 | 4.143e-05 0.000e+00 | | | | | | sum | 10511 10000 10000 | 4.143e-05 0.000e+00 | | | *------- End PYTHIA Event and Cross Section Statistics ----------------------------------------------------------* *------- VINCIA Statistics -------------------------------------------------------------------------------------* | | | | | Number of nonunity-weight events = none | | Number of negative-weight events = none | | | | weight(i) Avg Wt Avg Dev rms(dev) kUnwt Expected effUnw | | This run i = IsUnw <w> <w-1> 1/<w> Max Wt <w>/MaxWt | | User settings 0 yes 1.000 0.000 - 1.000 - - | | Var : VINCIA defaults 1 yes 1.000 0.000 - 1.000 1.000 1.000 | | Var : AlphaS-Hi 2 no 0.996 -3.89e-03 - 1.004 22.414 4.44e-02 | | Var : AlphaS-Lo 3 no 1.020 1.99e-02 - 0.981 43.099 2.37e-02 | | Var : Antennae-Hi 4 no 1.000 2.61e-04 - 1.000 5.417 0.185 | | Var : Antennae-Lo 5 no 0.996 -4.33e-03 - 1.004 10.753 9.26e-02 | | Var : NLO-Hi 6 yes 1.000 0.000 - 1.000 1.000 1.000 | | Var : NLO-Lo 7 yes 1.000 0.000 - 1.000 1.000 1.000 | | Var : Ordering-Stronger 8 no 1.004 4.48e-03 - 0.996 14.225 7.06e-02 | QCD | Var : Ordering-mDaughter 9 no 1.033 3.25e-02 - 0.968 55.954 1.85e-02 | | Var : Subleading-Color-Hi 10 no 1.001 7.37e-04 - 0.999 1.505 0.665 | Lecture | Var : Subleading-Color-Lo 11 no 1.006 6.44e-03 - 0.994 5.283 0.191 | III | | *------- End VINCIA Statistics ----------------------------------------------------------------------------------* P . Skands 43
Introduction to QCD 1. Fundamentals of QCD 2. Jets and Fixed-Order QCD 3. Monte Carlo Generators and Showers 4. Matching at LO and NLO 5. QCD in the Infrared Note: Teach-yourself PYTHIA tutorial posted at: www.cern.ch/skands/slides QCD Lecture III P . Skands 44
Supplementary Slides
Hard Processes Slide from T. Sjöstrand Wide spectrum from “general-purpose” to “one-issue”, see e.g. http://www.cedar.ac.uk/hepcode/ Free for all as long as Les-Houches-compliant output. I) General-purpose, leading-order: • MadGraph/MadEvent (amplitude-based, ≤ 7 outgoing partons): http://madgraph.physics.uiuc.edu/ • CompHEP/CalcHEP (matrix-elements-based, ∼≤ 4 outgoing partons) • Comix: part of SHERPA (Behrends-Giele recursion) • HELAC–PHEGAS (Dyson-Schwinger) QCD Lecture III P . Skands 46
Hard Processes Slide from T. Sjöstrand Wide spectrum from “general-purpose” to “one-issue”, see e.g. http://www.cedar.ac.uk/hepcode/ Free for all as long as Les-Houches-compliant output. I) General-purpose, leading-order: • MadGraph/MadEvent (amplitude-based, ≤ 7 outgoing partons): http://madgraph.physics.uiuc.edu/ • CompHEP/CalcHEP (matrix-elements-based, ∼≤ 4 outgoing partons) • Comix: part of SHERPA (Behrends-Giele recursion) • HELAC–PHEGAS (Dyson-Schwinger) II) Special processes, leading-order: • ALPGEN: W / Z+ ≤ 6j , n W + m Z + k H+ ≤ 3j , . . . • AcerMC: ttbb , . . . • VECBOS: W / Z+ ≤ 4j QCD Lecture III P . Skands 46
Hard Processes Slide from T. Sjöstrand Wide spectrum from “general-purpose” to “one-issue”, see e.g. http://www.cedar.ac.uk/hepcode/ Free for all as long as Les-Houches-compliant output. I) General-purpose, leading-order: • MadGraph/MadEvent (amplitude-based, ≤ 7 outgoing partons): http://madgraph.physics.uiuc.edu/ • CompHEP/CalcHEP (matrix-elements-based, ∼≤ 4 outgoing partons) • Comix: part of SHERPA (Behrends-Giele recursion) • HELAC–PHEGAS (Dyson-Schwinger) II) Special processes, leading-order: • ALPGEN: W / Z+ ≤ 6j , n W + m Z + k H+ ≤ 3j , . . . • AcerMC: ttbb , . . . • VECBOS: W / Z+ ≤ 4j III) Special processes, next-to-leading-order: Note: NLO codes not yet generally interfaced • MCFM: NLO W / Z+ ≤ 2j , WZ , WH , H+ ≤ 1j QCD to shower MCs • GRACE+Bases/Spring Lecture III P . Skands 46
Splitting Functions Altarelli-Parisi (E.g., PYTHIA) P (t 1 ,z 1 ) t 0 α abc � (t 2 .z 2 ) d P a = 2 π P a → bc ( z ) d t d z . b,c 1 + z 2 P q → qg ( z ) = C F 1 − z , (1 − z (1 − z )) 2 P g → gg ( z ) = N C , z (1 − z ) T R ( z 2 + (1 − z ) 2 ) , P g → qq ( z ) = 1 + z 2 e 2 P q → q � ( z ) = 1 − z , q 1 + z 2 e 2 P ⇥ → ⇥� ( z ) = 1 − z , ⇥ QCD Lecture III P . Skands 47
Splitting Functions Altarelli-Parisi (E.g., PYTHIA) P (t 1 ,z 1 ) t 0 α abc � (t 2 .z 2 ) d P a = 2 π P a → bc ( z ) d t d z . b,c 1 + z 2 P q → qg ( z ) = C F 1 − z , (1 − z (1 − z )) 2 P g → gg ( z ) = N C , z (1 − z ) T R ( z 2 + (1 − z ) 2 ) , P g → qq ( z ) = s 1 + z 2 e 2 P q → q � ( z ) = 1 − z , q 1 + z 2 e 2 P ⇥ → ⇥� ( z ) = 1 − z , ⇥ QCD Lecture III P . Skands 47
Splitting Functions Altarelli-Parisi (E.g., PYTHIA) P (t 1 ,z 1 ) t 0 α abc � (t 2 .z 2 ) d P a = 2 π P a → bc ( z ) d t d z . b,c 1 + z 2 P q → qg ( z ) = C F 1 − z , (1 − z (1 − z )) 2 P g → gg ( z ) = N C , i z (1 − z ) I T R ( z 2 + (1 − z ) 2 ) , P g → qq ( z ) = j s 1 + z 2 k e 2 K P q → q � ( z ) = 1 − z , q 1 + z 2 (s ij ,s jk ) (…) e 2 P ⇥ → ⇥� ( z ) = 1 − z , ⇥ (…) QCD Lecture III P . Skands 47
Splitting Functions ⇥ Altarelli-Parisi Dipole-Antennae (E.g., PYTHIA) (E.g., ARIADNE, VINCIA) P (t 1 ,z 1 ) t 0 d s ij d s jk α abc d P IK → ijk = 16 π 2 s a ( s ij , s jk ) � (t 2 .z 2 ) d P a = 2 π P a → bc ( z ) d t d z . b,c ⇤ 1 + z 2 q = 2 C F � 2 s ik s + s 2 ij + s 2 ⇥ a q ¯ q → qg ¯ P q → qg ( z ) = C F 1 − z , jk s ij s jk C A (1 − z (1 − z )) 2 � ⇥ 2 s ik s + s 2 ij + s 2 jk − s 3 a qg → qgg = ij P g → gg ( z ) = N C , s ij s jk i z (1 − z ) I C A � ⇥ T R ( z 2 + (1 − z ) 2 ) , 2 s ik s + s 2 ij + s 2 jk − s 3 ij − s 3 a gg → ggg = P g → qq ( z ) = j jk s ij s jk s 1 + z 2 q 0 q 0 = T R k e 2 K � ⇥ s − 2 s ij + 2 s 2 P q → q � ( z ) = 1 − z , a qg → q ¯ q ij s jk 1 + z 2 q 0 q 0 = a qg → q ¯ (s ij ,s jk ) (…) a gg → g ¯ e 2 q 0 q 0 P ⇥ → ⇥� ( z ) = 1 − z , ⇥ (…) … + non-singular terms QCD Lecture III P . Skands 47
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